I'm a gamer since 1990 or so. I owned a NES, SNES, N64, Game Cube, Wii, PS1, PS2, Xbox, Xbox 360, Dream Cast, Game Boy, Game Boy Color, Game Boy Advance, Virtual Boy, Game Boy SP, Nintendo DS, and 3DS.
Online games are great but sharing the same console with your friends or family has a different feel to it. Trash talk feels better, you get to spend time with friends and family rather than random online people and I have to emphasize that trash talk is approximately fifteen times better when done in person. For some reason, playing with friends makes bad games good and funny. In addition to that playing games online isn't necessarily a social activity. Sometimes it's as if you're playing with a bot. Split screen ensures social interactions... usually. So I'm going to go ahead and suggest a bunch of co-op friendly games from a few genres.
My top two picks in this area are New Super Mario Bros. for the Wii and Rayman Origins. They both play the pretty much the same and are both very good choices for a nice individual or co-op experience. I can almost guarantee a good time playing either one of them. Unfortunately the both share a similar flaw. In Super Mario sometimes your teammate can get in the way, taking away some valuable space you might need to stand in and in Rayman it's a little too easy to hit your mate by accidentally, knocking them towards their doom. Both games are quite forgiving to make up for it.
Now this is a tricky one. Most RPGs are either made for solo play or let you get a co-op partner that is a slightly glorified generic NPC follower. There are some that are made for co-op play such as Final Fantasy Crystal Chronicles and Zelda Four Swords. They were both designed with 2-4 players in mind. Another good choice would be Fable III. It's more like a regular RPG but your partner might feel a little left out when you're not in combat.
The Gears of War series and Army of Two are two great choices for some co-op action. They both encourage teamwork to some degree and I personally had a really good time playing both. However, Gears of War has the advantage of supporting around 10 times more modes than Army of Two does. Well, it's mostly hoard mode (survival against a bunch of monster waves) and different types of fighting.
Guitar Hero and Rock Band. Playing with a full band is far more entertaining than a solo act and I don't think I need to delve any deeper into explaining this.
Note: Band members need to understand that drummers get tired sometimes on higher difficulties...